Find the domain of each of the following function: `f(x)=2^(sin^(-1)x)+1/(sqrt(x-2))`

Explanation

To find the domain of the function \( f(x) = 2^{\sin^{-1} x} + \frac{1}{\sqrt{x-2}} \), we need to consider the domains of each component of the function separately and then find their intersection.

### Step 1: Determine the domain of \( 2^{\sin^{-1} x} \)

The function \( \sin^{-1} x \) (the inverse sine function) is defined for \( x \) in the interval \([-1, 1]\). Therefore, the domain of \( 2^{\sin^{-1} x} \) is:

\[ D_1 = [-1, 1] \]

### Step 2: Determine the domain of \( \frac{1}{\sqrt{x-2}} \)

The expression \( \sqrt{x-2} \) is defined when \( x - 2 > 0 \), which simplifies to:

\[ x > 2 \]

Since we are dealing with a fraction, we also need to ensure that the denominator is not zero. Thus, the domain for this part is:

\[ D_2 = (2, \infty) \]

### Step 3: Find the intersection of the two domains

Now, we need to find the intersection of the two domains \( D_1 \) and \( D_2 \):

- \( D_1 = [-1, 1] \)
- \( D_2 = (2, \infty) \)

The intersection of these two sets is empty, as there are no values of \( x \) that can satisfy both conditions simultaneously. Therefore, the domain of the function \( f(x) \) is:

\[ D = D_1 \cap D_2 = \emptyset \]

### Conclusion

The domain of the function \( f(x) = 2^{\sin^{-1} x} + \frac{1}{\sqrt{x-2}} \) is:

\[ \text{Domain } f(x) = \emptyset \]